The Integrable Systems group has its own seminar series and frequently hosts Workshops and Conferences in Leeds. It has national and international links and collaborative networks. Some of our previous PhD students and post-doctoral researchers contributed lectures in the LMS Classical and Quantum Integrability programme.

Below is a selection of PhD topics proposed in our group. For more information about these, feel free to contact the designated supervisor directly.

Prospective students are welcome to e-mail any staff member of the group to discuss research possibilities. Further details of possible funding and of how to apply can be found on the Postgraduate research opportunities page.

**Description**: The project deals with a novel quantum theory that is based on the mathematical structure behind so-called integrable systems, i.e., models that on the classical level are described by certain (mostly nonlinear) differential or difference equations and that exhibit the aspect of `exact solvability’ – the fact that they are amenable to exact (rather than approximate or numerical) methods for their solution. Such systems possess a rich mathematical structure (e.g. connections to infinite-dimensional Lie algebras and quantum groups). A key feature is the notion of multidimensional consistency, the property that such models can be extended to compatible systems of equations in spaces of arbitrary dimension. In 2009 Lobb and Nijhoff proposed a novel variational (i.e., least-action) principle that describes this remarkable feature within a Lagrangian formalism.

The new principle forms potentially a paradigm for a new type of fundamental physics.

Recently, the ideas behind this new approach were extended to the quantum realm, and the first steps were taken to formulate a quantum version of this variational principle. The project for a PhD student is to continue this work into the quantum variational principle, elaborating various (integrable) model systems aimed at building a coherent framework together with the development of some new mathematical methodologies.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and entertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.

**Description**: The term, Integrable Systems refers to a wide class of very special models, described by nonlinear differential (in the continuous case) or difference (in the discrete case) equations possessing a number of remarkable properties. One of the outstanding features is that these equations are exactly solvable in the sense that, rather than having to rely on numerical techniques or approximations, these equations allow for exact (albeit highly nontrivial) methods for their solution. Examples of these are the well-known soliton solutions of certain partial differential equations in this class. In the discrete case, the theory behind these model difference equations has been steadily developing, mostly from the early 1990s onward, together with the mathematical theories which had to be developed alongside (as they were largely non-existent in the discrete case).

The project focuses on elliptic integrable systems, which are those cases where coefficients and generating quantities for these equations are given in terms of elliptic functions. The latter generalizations of trigonometric functions have a rich mathematical structure, some aspects of which are still being explored (e.g. they play a role in Fermat’s last theorem), and the integrable models which are defined through them are in a sense at the top of the food chain of models: they form the richest and most general class of equations. To a large extent the solution structure of those elliptic models has yet to be unravelled, and that will form the core of the project. In the project the student will investigate specific examples of such elliptic discrete integrable systems, which will entail not only to try and apply some well-tested techniques to these more complex cases for finding explicit solutions, but also to develop some methods for generating novel examples of such systems. As motivation, these models are expected to have relevance not only for creating novel mathematics, but also potentially for finding new models of fundamental physics.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and entertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.